Numerical results: The numerical results are obtained by a direct solution of the non-Hermitian Schrodinger equation. As shown above in this case we have \(\left[S(t)\right]_{t\rightarrow \infty }=0\)  and we always eventually detect the particle.
In Fig. (REF ) (left panel) we plot the
decay of survival probability on a system for which \(N=15\)  for three different values of \(w\) . The survival
probability cascades to 0 over time. We observe a non-monotonic dependence with \(S(t)\)  decaying slowly for both very small \(w=0.1\)  as well large \(w=5\) . The several plateaues in the survival probability can be understood as arising from ballistic propagation of the particle with a
a group velocity \(\approx 2\) , such that as the wave packet hits the detector, the surival probability plunges considerably. The velocity 2 corresponds to the maximum group velocity \(d\epsilon _q/dq\) .
In Fig. (REF ) (right panel) we show the decay of survival probability
for relatively larger systems. In all cases, \(\psi _{i}(0)=\delta _{i,20}\) 
and \(w=2.0\) . In this case the first short plateau in \(S(t)\)  corresponds to the time taken for the wave-packet to reach the detector while the second longer plateau corresponds to the wave-packet traveling from the origin to the right end and back. Hence this second time-scale can be seen to be proprtional to the lattice size \(N\) . The dashed line indicates the infinite time survival probability \(S_{\mathbb {N}}(w,20)\)  on the semi-infinite lattice \({\mathbb {N}}\)  which is computed exactly in the next section (see Eq. (REF )). Similar finite size effects have been discussed recently in Ref. [1]}.
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